Lesson Background and Concepts for Teachers
This lesson and its associated activity is suitable for the end of the first semester of the school year for high school AP Calculus courses, serving as a major grade for the last six-week period with the results presentation-report (see the associated activity) taken as the first-semester finals test.
Provided in this section is the detailed process to obtain the formulas to predict velocities and friction forces for a spherical body rolling along a curved path. Parts 1 and 2 contain basic concepts students usually cover in a pre-
course. Parts 3 and 4 are extensions of these concepts—and the core of this lesson. The same information is provided in
as a 10-page student handout.
1.0 Basic Concepts
1.1 Rotational Movement Kinematics
=s/r (Figure 2), (
= circle radius,
= arc length),
s = θ * r
, then the angular and tangential velocities are related as:
If angular acceleration is defined as: α = dω/dt, and tangential acceleration as: a
The angular and tangential accelerations are then related as: a
= α * r.
1.2 Rotational Kinetic Energy and Moment of Inertia of a Rigid Body
For a single particle, the kinetic energy for linear movement is defined as
K = ½ mv
. Similarly, for a single particle, the kinetic energy for rotational movement is given by the formula
K = ½ Iω
I = mr
is known as moment of inertia of the particle, and
is the distance from the point of rotation.
For a system of particles, its moment of inertia is the sum of the individual moments
I = ∑m
. This definition can be extended to compute the moment of inertia for a continuous rigid body:
For the special case analyzed in this project, the moment of inertia for a solid homogeneous sphere is:
1.3 Angular Momentum and Torque of a Rigid Body
From the law of lever, the torque, or amount of rotation produced by a force, is mathematically defined as
is the perpendicular force applied on the lever at a distance
from the fulcrum. Extending this definition, taking the force
as a vector
non-perpendicular to the vector
that gives the position of the point where the force is applied with respect to the rotation point, the torque is defined as the vector product:
where the torque
is represented as a vector perpendicular to the plane (Figure 3) defined by vectors
. The magnitude of this vector is given by the product of the magnitudes of the vectors
) times the sine of the angle between them:
When the angle is 90
are perpendicular, and the original law of lever is obtained.
For a single particle, the linear momentum is defined as the vector:
= m * v
, and by definition, the force or Newton’s second law for a single particle when its mass changes with the time is given by:
When the mass of the particle is constant:
Similar to the linear momentum, the angular momentum of a particle is defined as:
and similar to the force definition for linear movement in terms of the linear momentum, the force producing a rotational movement, or torque, is defined as:
For a particle of constant mass in a circular movement (
, the tangential velocity (θ = 90
), the magnitude of the torque is given by:
a formula similar to Newton’s second law for a rotating rigid body with constant mass.
2. Force of Friction for a Spherical Rigid Body Rolling without Slipping on an Incline Surface
A spherical rigid body of mass
is rolling down an incline surface. The forces producing this movement are the weight of the body
and the force of friction
Because the body is rolling instead sliding, the friction to consider in this problem is static friction. So, the coefficient to use in the friction force calculations is the static friction coefficient
The body rolls because a torque is produced by the friction force
and the component of the body’s weight parallel to the inclined surface (Figure 4). All forces on the rolling body can be analyzed by using a free-body diagram.
The forces that make this spherical solid of radius
roll down the incline plane are those along the
. Using Newton’s second law to describe the solids linear movement:
The force that makes the solid rotate is the torque produced by the friction force:
= sphere’s momentum of inertia,
= sphere’s angular acceleration,
= sphere’s radius for a homogenous spherical object:
I = (2/5)*m*r
. Substituting this value, and
α = a/r
The force of friction can be expressed as:
Substituting (3) in (1):
Substituting the obtained acceleration value in (3), the force of friction can be expressed in terms of the sphere’s weight and the angle of the incline:
By definition, the static friction force is proportional to the normal force, or force the surface applies on the sphere to balance the component of the sphere’s weight perpendicular to the surface.
is the static friction coefficient and
is the magnitude of the normal force. For this problem, the value of the normal force can be found in the free-body diagram:
Substituting this value in (5):
Combining equations (4) and (6), an expression for the coefficient of static friction can be found:
Equation (7) states that for a spherical body rolling on an incline, the coefficient of static friction is a function of the inclined surface angle
, specifically, of the tangent of this angle or slope of the inclined surface.
3. Force of Friction for a Spherical Rigid Body Rolling without Slipping on a Variable Slope Path
A sphere rolling on a path with variable slope can be visualized as rolling on a sequence of inclined tangent straight surfaces. If the path is given as a function
y = f (x)
, differentiable at every point, the slope of any these inclines, defined as
m = tan θ
(Figure 5), can be found using the derivative
Using the above expression in (7), the coefficient of static friction for a sphere rolling on a variable slope path is given by:
Expression (8) states that the static friction coefficient varies from point to point on the path. Using equation (8) in (6), the static friction force for a rolling sphere of mass
Using trigonometry, it is possible to express cos
also in terms of
. Because tan
θ = f ’(x)
Solving with right triangles (Figure 6):
Using this expression for cos
Equations (8) and (10) seem to be adequate to model the problem of a sphere rolling on a variable slope path with friction, but something needs to be fixed. Both equations give negative values for some sections of the path, and by definition, the static friction coefficient is always positive.
The problem comes from the sign of the derivative and the possibility of negative slopes. For the inclined plane, the elevation angle is always positive. Because in this problem we approached the path with variable slope as a sequence of inclines, the slope given by
must always be positive. With this restriction, equations (8) and (10) must be given in terms of the absolute value of
4. Work-Energy Theorem for a Sphere Rolling on a Variable Slope Path with Friction
The work-energy theorem states that the mechanical energy (kinetic energy + potential energy) of an isolated system under only conservative forces remains constant.
is the mechanical energy of the system,
is the kinetic energy
is the potential energy
, and the indexes
indicate the energies at the end and beginning of the process, respectively.
While energy cannot be created nor destroyed in an isolated system, it can be internally converted to other energy forms.
When a non-conservative force like friction is considered in a system, the work-energy theorem states that the work done by the non-conservative forces,
, is equivalent to the change in mechanical energy:
Under non-conservative forces,
is no longer zero, but another key difference exists. Meanwhile for conservative systems, the work done by conservative forces depends only on the initial and final positions. For non-conservative systems, the work done by non-conservative forces, like friction, depends on the path or trajectory, or on the time these forces affect the system. In a system under conservative forces, the work on a closed loop is always zero, while in a system under non-conservative forces it is not.
By definition, mechanical work is the product of the displacement times the force component along the displacement (Figure 7):
For the variable slope path
y = f (x)
, the work done by the force of friction (12) is:
Taking differential displacements along the path:
If the differential arc
is defined as:
can be set as a function of
Substituting in (15) in (14):
|dx| = dx
Because the friction force always acts against the movement, the work done by friction is always negative. Then:
Again taking small displacement
along the path:
For the entire path, the total work is the sum of the work done along the little displacements the path had been divided into (Figure 8).
Substituting equation (16) in (13):
because the height of the object on the path is given by the value of function
h = f (x)
Using the above equation, it is possible to find the velocity of the rolling body at the end of every little displacement (Figure 8), knowing the values of its velocity at the beginning of the displacement, and of the corresponding potential energy change:
The purpose of this lesson is to provide the necessary physics background so students understand why equation (18) is useful in the design of a roller coaster, which they need in order to complete the associated activity,
Mathematically Designing a
. If time is limited, you may want to skip Sections 1 and 2, and assign students, as homework, to watch the online tutorial videos about the topics of friction, and rotational dynamics (S1). Be sure students fully understand these concepts, or it will be difficult for them to understand the next ones.
Since no online tutorials are available for Sections 3 and 4, which are the core lesson concepts, they must be covered by the teacher in class. Use the
A Tale of Friction Presentation
to help in the explanation of these important concepts, and hand out the
. Having these 10 pages of notes in hand frees students to pay more attention to the teacher explanations. Inform students of the importance of these concepts to completing the hands-on activity they will engage in later, and alert them that you will lecture them as a college professor would. As makes sense for your students, verify they understand what you are explaining. The
are provided as one way to assess student comprehension.
Become familiar with all the concepts and mathematical steps in this lesson. Feel free to contact the author for clarification of any of the above depicted steps.
suggested sequence to conduct the lesson
(optional) Basic Concepts
Section 1.1: Rotational Movement Kinematics
Section 1.2: Rotational Kinetic Energy and Moment of Inertia of a Rigid Body
Section 1.3: Angular Momentum and Torque of a Rigid Body
Section 2: Force of Friction for a Spherical Rigid Body Rolling without Slipping on an Incline Surface
Rolling with Friction A
Section 3: Force of Friction for a Spherical Rigid Body Rolling without Slipping on a Variable Slope Path
Rolling with Friction B
Section 4: Work-Energy Theorem for a Sphere Rolling on a Variable Slope Path with Friction
The change in the velocity of an object. The average acceleration is defined as the change in velocity divided by time. The instantaneous acceleration is defined as the acceleration at any particular time period. In calculus, given the velocity of a body as a function of the time, the instantaneous acceleration is the derivative if the velocity with respect to the time.
In a very basic approach, a function whose graph does not have any void and/or it is not broken, that is, can be drawn without lifting the pencil from the paper. More formally, a function is continuous at a point in its domain if a sufficiently small change in the input results in an arbitrarily small change in the output. In calculus, a function is continuous at a point x = c if and only if all the next three conditions are met: 1) the function is defined at x = c, 2) the limit of the function at x = c exists, and 3) the limit and the value of the function at x = c are equal.
The limit of the ratio of the change in a function to the corresponding change in its independent variable as the latter change approaches zero. Geometrically, this rate of change gives the slope of the tangent line at a point on the function’s graph. The graph of the function at that point must be continuous and smooth, that is, the function cannot have a peak at this point.
differentiable function :
A function whose derivative exists at each point in its domain. Geometrically, the function whose graph is continuous and smooth such that a tangent line exists for every point on the graph.
A principle that states that in a system that does not undergo any force from outside the system, the amount of energy is constant, irrespective of its changes in form.
The resistance to motion of one object moving relative to another. The surface resistance to relative motion, as of a body sliding or rolling. The force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other.
The force of attraction between masses; the attraction of the earth’s mass for bodies near its surface. The gravitational force between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them.
A force that resists a change in velocity of an object. It is equal to the applied force, but with opposite direction. Because of this force, every object in a state of uniform motion, or rest, tends to remain in that state unless an external force is applied to it.
The energy possessed by an object due to its motion or movement. It is proportional to its mass and to the square of its velocity. The magnitude of this energy arises from the net work done on the object, to accelerate it from rest to its final velocity. Kinetic energy can be transformed again in work.
A force that acts between moving surfaces. It is also known as sliding friction or moving friction, and it is the amount of retarding force between two objects that are moving relative to each other.
maximum of a function:
The largest value of a function, either within a given range or on the entire domain. Formally, a function f (x) has a maximum at x = c, a f (x), for a ≤ x ≤ b.
The capacity of a physical system to change an object’s state of motion.
The amount of energy transferred by a force that moves an object. The work is calculated by multiplying the applied force by the amount of movement of the object.
minimum of a function:
The smallest value of a function, either within a given range or on the entire domain. Formally, a function f (x) has a minimum at x = c, a
A conic section formed by the intersection of a vertical cone by a plane parallel to the cone’s side. A curve where any point is at an equal distance from a fixed point, the focus, and a fixed straight line, the directrix.
The point where a parabola crosses its axis of symmetry. It is the maximum point when the parabola opens downwards, or the minimum point if the parabola opens upwards.
A function that is defined by multiple sub-functions, each sub-function applying to a certain interval of the main function’s domain. Also known as a hybrid function.
The energy possessed by a body by virtue of its position relative to others. In a gravitational field, potential energy is the energy stored in an object as the result of its vertical position or height. The energy is stored as the result of the gravitational attraction of the earth for the object.
An amusement park ride that consists on an elevated railroad track designed with sharp curves and steep slopes on which people move in small, fast and open rolling cars.
The oldest roller coaster design and a predecessor to the modern day roller coaster. Descended from Russian winter sled rides on hills of ice, these early roller coasters were open wheeled carts or open train carts on tracks of elevated up-hills and down-hills supported by wooden or steel structures.
The straight line joining two points on a curve.
The friction that exists between a stationary object and the surface on which it rests—the force that keeps an object at rest and that must be overcome to start moving an object. Pushing horizontally with a small force, static friction establishes an equal and opposite force that keeps the object at rest.
The straight line that touches a curve at a point without crossing over.
The time rate of change of position of a body in a specified direction. The average velocity is defined as the change in position divided by the time of travel. The instantaneous velocity is simply the average velocity at a specific instant in time. In calculus, given the position of a body as a function of the time, the instantaneous velocity is the derivative of the position with respect to the time.
A principle that states that the work done by all forces acting on a particle equals the change in the particle’s kinetic energy.